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Mirrors > Home > ILE Home > Th. List > cnvss | GIF version |
Description: Subset theorem for converse. (Contributed by NM, 22-Mar-1998.) |
Ref | Expression |
---|---|
cnvss | ⊢ (𝐴 ⊆ 𝐵 → ◡𝐴 ⊆ ◡𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel 3149 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → (〈𝑦, 𝑥〉 ∈ 𝐴 → 〈𝑦, 𝑥〉 ∈ 𝐵)) | |
2 | df-br 4003 | . . . 4 ⊢ (𝑦𝐴𝑥 ↔ 〈𝑦, 𝑥〉 ∈ 𝐴) | |
3 | df-br 4003 | . . . 4 ⊢ (𝑦𝐵𝑥 ↔ 〈𝑦, 𝑥〉 ∈ 𝐵) | |
4 | 1, 2, 3 | 3imtr4g 205 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝑦𝐴𝑥 → 𝑦𝐵𝑥)) |
5 | 4 | ssopab2dv 4277 | . 2 ⊢ (𝐴 ⊆ 𝐵 → {〈𝑥, 𝑦〉 ∣ 𝑦𝐴𝑥} ⊆ {〈𝑥, 𝑦〉 ∣ 𝑦𝐵𝑥}) |
6 | df-cnv 4633 | . 2 ⊢ ◡𝐴 = {〈𝑥, 𝑦〉 ∣ 𝑦𝐴𝑥} | |
7 | df-cnv 4633 | . 2 ⊢ ◡𝐵 = {〈𝑥, 𝑦〉 ∣ 𝑦𝐵𝑥} | |
8 | 5, 6, 7 | 3sstr4g 3198 | 1 ⊢ (𝐴 ⊆ 𝐵 → ◡𝐴 ⊆ ◡𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2148 ⊆ wss 3129 〈cop 3595 class class class wbr 4002 {copab 4062 ◡ccnv 4624 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-in 3135 df-ss 3142 df-br 4003 df-opab 4064 df-cnv 4633 |
This theorem is referenced by: cnveq 4799 rnss 4855 relcnvtr 5146 funss 5233 funcnvuni 5283 funres11 5286 funcnvres 5287 foimacnv 5477 tposss 6243 structcnvcnv 12469 pw1nct 14603 |
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